Mixtures & Alligation
🔒 Log in to trackAlligation is the highest-leverage tool in arithmetic — it solves mixtures, dilutions, adulteration profits and even average-group questions in one crossing. Replacement chains and two-mixture blending complete the set. Expect one sure question per Tier 1 shift and more in Tier 2.
One page per subtopic: detailed notes, every question type, formulas, tricks and practice sets.
Every formula on one printable page, grouped by subtopic.
5 exam-level questions worked step by step.
66 questions — untimed practice or a timed test with analysis.
Track record in the exam
Questions per shift in recent SSC CGL papers.
Test difficulty mix (66 questions)
Question patterns exams keep repeating
Taken from previous-year papers. If a pattern is marked "very common", expect to see it in your exam.
Mixing two prices
very commonTwo prices are mixed to get a target price. The ratio is asked.
Subtract the target from each price, then cross over.
The target must lie between the two prices.
In what ratio should tea at ₹320/kg be mixed with tea at ₹250/kg to get a mix at ₹300/kg?
Dearer tea: 320 − 300 = 20 away
Cheaper tea: 300 − 250 = 50 away
Cross over: dearer : cheaper = 50 : 20 = 5 : 2
Drawing out and replacing
very commonSome litres are drawn out and replaced with water, once or more.
Each time, the same fraction of milk stays.
Multiply by that fraction once per turn.
A vessel has 80 litres of milk. 8 litres are drawn and replaced by water. This is done once more. How much milk is left?
Each time 8 ÷ 80 = 1/10 goes, so 9/10 stays
After 1st: 80 × 0.9 = 72
After 2nd: 72 × 0.9 = 64.8 litres
Adding water
very common"How much water must be added to get a stated strength?"
The milk does not change.
Find the new total from the milk and the percentage.
How much water must be added to 60 litres of milk so milk is 75% of the mixture?
Milk stays 60 L and is 75% of the total
Total = 60 ÷ 75 × 100 = 80 L
Water added = 80 − 60 = 20 litres
Milk and water ratio
commonA ratio and a total are given. Find the amounts or a new ratio.
Split the total by the ratio.
Add to one part only, and keep the other part fixed.
A 40-litre mixture has milk and water in ratio 3 : 1. How much milk must be added to make it 4 : 1?
Milk = 30 L, water = 10 L
Water stays 10 L, so milk must be 4 × 10 = 40 L
Milk to add = 40 − 30 = 10 litres
Profit from adding water
commonA seller adds water (or a cheap item) and sells at the full price.
Water costs nothing but is sold at milk's price.
Gain % = water ÷ milk × 100.
A milkman adds water equal to one-fifth of the milk and sells the mixture at the cost price of milk. Find his gain %.
Take 5 L milk + 1 L water = 6 L
Cost = 5, sale = 6
Gain = 1 on 5 = 20%
Mixing two mixtures
commonTwo ready-made mixtures are combined. The new ratio is asked.
Take the same amount of each, like 10 litres.
Add up the milk and water separately.
Milk : water is 4 : 1 in one vessel and 3 : 2 in another. They are mixed in equal amounts. Find the new ratio.
Take 10 L of each
Vessel 1: milk 8, water 2. Vessel 2: milk 6, water 4
Total milk 14, water 6 → 14 : 6 = 7 : 3
Mixing averages
commonTwo groups have different averages. The overall average is given.
Subtract each average from the overall average, then cross over.
The overall average must lie between the two.
Boys average 70 marks and girls average 75. The class average is 72. Find boys : girls.
Boys: 72 − 70 = 2 away
Girls: 75 − 72 = 3 away
Cross over: boys : girls = 3 : 2
Check: (3 × 70 + 2 × 75) ÷ 5 = 72, so the ratio is 3 : 2
Cheating with water
commonWater is a % of the mixture and it is sold at milk's price.
Water is sold at milk's price but costs nothing.
Gain % = water ÷ milk × 100.
A milkman's mixture is 20% water. He sells it at the cost price of milk. Find his gain %.
Take 100 L mixture: 20 L water, 80 L milk
Cost = 80, sale = 100
Gain = 20 on 80 = 25%